Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras

TL;DR

This paper introduces and classifies Cartan-Weyl 3-algebras, extending the Lie algebra root space decomposition to 3-algebras, with potential applications in generalized symmetries and BLG theory.

Contribution

It provides a complete classification of Cartan-Weyl 3-algebras, generalizing the Cartan-Weyl basis concept from Lie algebras to 3-algebras.

Findings

Many known metric Lie 3-algebras are special cases of Cartan-Weyl 3-algebras.

The structure of Cartan-Weyl 3-algebras is elegant and simple.

Potential applications in describing generalized symmetries and BLG theory.

Abstract

As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^\alpha that are characterized by a root space of non-degenerate one-forms \alpha. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.